Given ΔGHI with G(4, –3), H(–4, 2), and I(2, 4), find the perpendicular bisector of in standard form.
To find the equation of the perpendicular bisector of ΔGHI, we need to find the midpoint and the slope of the line segment GH.
The midpoint of GH is the average of the x-coordinates and the average of the y-coordinates:
Midpoint of GH = ((4 - 4)/2, (-3 + 2)/2) = (0, -0.5).
The slope of the line segment GH is the change in y divided by the change in x:
Slope of GH = (-3 - 2)/(4 - (-4)) = -5/8.
The slope of the perpendicular bisector of GH is the negative reciprocal of the slope of GH, which is 8/5.
Using the point-slope form y - y₁ = m(x - x₁), where (x₁, y₁) = (0, -0.5) and m = 8/5, we can write the equation of the perpendicular bisector in point-slope form:
y - (-0.5) = (8/5)(x - 0).
Simplifying, we get:
y + 0.5 = (8/5)x.
To write the equation in standard form, we need to multiply both sides of the equation by 5 to get rid of the fraction:
5y + 2.5 = 8x.
Rearranging the equation, we get:
8x - 5y = -2.5.
Therefore, the equation of the perpendicular bisector of ΔGHI in standard form is:
8x - 5y = -2.5.